I would like to make a 2 players app game, where 2 players can compete with each other in Real Time.

If I want the player to be able to compete with the public (anyone who registered with my game able to compete with one another), do I need a real time server?

If I want the player to be only able to compete with his facebook friends, do I need a real time server?

Do I need a server at all in the first place, for 2 players game? What else do i need? Eg, networking coding?

Please let me know. Thanks!!

You would likely need a server to host matchmaking between players - regardless of if this is from your complete player pool or just facebook.

For just two players connecting to each other, you can do direct connections or maybe host a NAT punchthrough intermediary server. You shouldn't need a heavy-duty server for that.

You will absolutely need to be learning some network coding to make this happen. This forum's FAQ is a good resource.